Large void zones and occupation times for coalescing random walks

The basic coalescing random walk is a system of interacting particles. These particles start from every site of Zd, and each moves independently as a continuous-time random walk. When two particles visit the same site, they coalesce into a single particle. We are interested in: (a) the radius Rd(T) of the largest ball centered at the origin which does not contain any particle at time T and (b) the amount of time Λd(T) when the origin is occupied during [0,T]. We describe the almost sure asymptotic behaviours of Rd(T) and Λd(T) (when T→∞), in three different regimes depending on whether d=1, d=2 or d⩾3.